We provide Schauder estimates for nonlinear Beltrami equations and lower bounds of the Jacobians for homeomorphic solutions. The results were announced in [1] but here we give detailed proofs.
Keywords: Quasiconformal mappings, Nonlinear Beltrami equation, Schauder estimates, Non-vanishing of the Jacobian
@article{AIHPC_2017__34_6_1543_0,
author = {Astala, Kari and Clop, Albert and Faraco, Daniel and J\"a\"askel\"ainen, Jarmo and Koski, Aleksis},
title = {Nonlinear {Beltrami} operators, {Schauder} estimates and bounds for the {Jacobian}},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1543--1559},
year = {2017},
publisher = {Elsevier},
volume = {34},
number = {6},
doi = {10.1016/j.anihpc.2016.10.008},
zbl = {1375.30024},
mrnumber = {3712010},
language = {en},
url = {https://www.numdam.org/articles/10.1016/j.anihpc.2016.10.008/}
}
TY - JOUR AU - Astala, Kari AU - Clop, Albert AU - Faraco, Daniel AU - Jääskeläinen, Jarmo AU - Koski, Aleksis TI - Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 1543 EP - 1559 VL - 34 IS - 6 PB - Elsevier UR - https://www.numdam.org/articles/10.1016/j.anihpc.2016.10.008/ DO - 10.1016/j.anihpc.2016.10.008 LA - en ID - AIHPC_2017__34_6_1543_0 ER -
%0 Journal Article %A Astala, Kari %A Clop, Albert %A Faraco, Daniel %A Jääskeläinen, Jarmo %A Koski, Aleksis %T Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 1543-1559 %V 34 %N 6 %I Elsevier %U https://www.numdam.org/articles/10.1016/j.anihpc.2016.10.008/ %R 10.1016/j.anihpc.2016.10.008 %G en %F AIHPC_2017__34_6_1543_0
Astala, Kari; Clop, Albert; Faraco, Daniel; Jääskeläinen, Jarmo; Koski, Aleksis. Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 6, pp. 1543-1559. doi: 10.1016/j.anihpc.2016.10.008
[1] Manifolds of quasiconformal mappings and the nonlinear Beltrami equation | arXiv | Zbl | DOI
[2] Uniqueness of normalized homeomorphic solutions to nonlinear Beltrami equations, Int. Math. Res. Not., Volume 2012 (2012) no. 18, pp. 4101–4119 | Zbl | MR | DOI
[3] Quasiregular mappings and Young measures, Proc. R. Soc. Edinb., Sect. A, Volume 132 (2002) no. 5, pp. 1045–1056 | Zbl | MR | DOI
[4] Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009 | Zbl | MR
[5] Beltrami operators in the plane, Duke Math. J., Volume 107 (2001) no. 1, pp. 27–56 | Zbl | MR | DOI
[6] Symposia Mathematica, Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, Volume vol. XVIII, Academic Press, London (1976), pp. 485–499 (INDAM, Rome, 1974) | Zbl | MR
[7] G-closed classes of elliptic operators in the complex plane, Ric. Mat., Volume 54 (2005) no. 2, pp. 403–432 | Zbl | MR
[8] Quasiconformal mappings and non-linear elliptic equations in two variables. I, II, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys., Volume 22 (1974), pp. 473–478 (479–484) | Zbl | MR
[9] Weighted estimates for Beltrami equations, Ann. Acad. Sci. Fenn., Math., Volume 38 (2013) no. 1, pp. 91–113 | Zbl | MR
[10] Tartar conjecture and Beltrami operators, Mich. Math. J., Volume 52 (2004) no. 1, pp. 83–104 | Zbl | MR | DOI
[11] Compactness versus regularity in the calculus of variations, Discrete Contin. Dyn. Syst., Ser. B, Volume 17 (2012) no. 2, pp. 473–485 | Zbl | MR
[12] Tartar's conjecture and localization of the quasiconvex hull in , Acta Math., Volume 200 (2008), pp. 279–305 | Zbl | MR | DOI
[13] Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983 | Zbl | MR
[14] Symposia Mathematica, Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, Volume vol. XVIII, Academic Press, London (1976), pp. 501–517 (INDAM, Rome, 1974) | Zbl | MR
[15] Hardy–Littlewood inequality for quasiregular mappings in certain domains in , Ann. Acad. Sci. Fenn., Ser. A 1 Math., Volume 10 (1985), pp. 267–282 | Zbl | MR
[16] Singular Integrals and Beltrami Type Operators in the Plane and Beyond, Department of Mathematics and Statistics, University of Helsinki, 2011 (Master's thesis)
[17] Universal potential estimates, J. Funct. Anal., Volume 262 (2012) no. 10, pp. 4205–4269 | Zbl | MR | DOI
[18] Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968 | MR
[19] Normal families of quasimeromorphic mappings, Proc. Am. Math. Soc., Volume 84 (1982), pp. 35–43 | Zbl | MR | DOI
[20] On the solutions of quasi-linear elliptic partial differential equations, Trans. Am. Math. Soc., Volume 43 (1938) no. 1, pp. 126–166 | JFM | MR
[21] Elliptic Systems and Quasiconformal Mappings, John Wiley and Sons, New York, 1988 | MR
[22] Über lineare elliptische Differentialgleichungen zweiter Ordnung, Math. Z., Volume 38 (1934) no. 1, pp. 257–282 | Zbl | MR | JFM | DOI
[23] Numerische Abschätzungen in elliptischen linearen Differentialgleichungen, Stud. Math., Volume 5 (1935), pp. 34–42 | Zbl | JFM
[24] On Tartar's conjecture, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 10 (1993), pp. 405–412 | Zbl | MR | Numdam | DOI
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